The Lebesgue Constant for the Periodic Franklin System

Details The Lebesgue Constant for the Periodic Franklin System The Lebesgue Constant for the Periodic Franklin System

2011-03-10

arxiv.org/abs/1103.1950v3

Mathematics

Numerical Analysis

Mathematica Notebook for creating Table 1 on page 21 is attached

Scientific paper

We identify the torus with the unit interval $[0,1)$ and let $n,\nu\in\mathbb{N}$, $1\leq \nu\leq n-1$ and $N:=n+\nu$. Then we define the (partially equally spaced) knots \[ t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{for}j=0,...,2\nu, \frac{j-\nu}{n}, & \text{for}j=2\nu+1,...,N-1.] Furthermore, given $n,\nu$ we let $V_{n,\nu}$ be the space of piecewise linear continuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$. Finally, let $P_{n,\nu}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,\nu}.$ The main result is \[\lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=2+\frac{33-18\sqrt{3}}{13}.\] This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is $2+\frac{33-18\sqrt{3}}{13}$.

Affiliated with

Also associated with

No associations

Rating

The Lebesgue Constant for the Periodic Franklin System does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Lebesgue Constant for the Periodic Franklin System, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Lebesgue Constant for the Periodic Franklin System will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-630490